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EXAMPLE
| E.g.f. A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2073*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(2*n) for n>=1, k>=0:
exp(x)*A(x)^2: [(1), 3, 17, 219, 5665, 239283, 14432433, ...];
exp(x)*A(x)^4: [1,(5), 41, 605, 15633, 638325, 37250233, ...];
exp(x)*A(x)^6: [1, 7,(73), 1207, 31825, 1274407, 72322201, ...];
exp(x)*A(x)^8: [1, 9, 113,(2073), 56545, 2249769, 124959057, ...];
exp(x)*A(x)^10:[1, 11, 161, 3251,(92481), 3695451, 202282081, ...];
exp(x)*A(x)^12:[1, 13, 217, 4789, 142705,(5775133), 313637833, ...];
exp(x)*A(x)^14:[1, 15, 281, 6735, 210673, 8688975,(471058953), ...]; ...
then the terms along the main diagonal form this sequence shift left.
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PROG
| (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(n=0, n, A=exp(serreverse(x/A^2))); n!*polcoeff(A, n)}
{a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(2*n), n-1))}
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